Publications

We prove a strongly polynomial bound on the circuit diameter of polyhedra, resolving the circuit analogue of the polynomial Hirsch conjecture. Specifically, we show that the circuit diameter of a polyhedron $P = \{x \in \mathbb{R}^n : Ax = b, x \geq 0\}$ with $A \in \mathbb{R}^{m \times n}$ is $O(m^2 \log m)$. Our construction yields monotone circuit walks, giving the same bound for the monotone circuit diameter.
The circuit diameter, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015), is a natural relaxation of the combinatorial diameter that allows steps along circuit directions rather than only along edges. All prior upper bounds on the circuit diameter were only weakly polynomial. Finding a circuit augmentation algorithm that matches this bound would yield a strongly polynomial time algorithm for linear programming, resolving Smale's 9th problem.

We present improved running time and iteration complexities of interior point methods for linear programs parametrized by the straight line complexity, i.e., the minimum number of segments of any piecewise linear curve traversing a particular neighborhood of the central path. While the standard measure of progress is the reduction in duality gap, the straight line complexity provides a stronger instance-wise bound, reflecting the combinatorial structure of the problem.
Our first main result is a wide neighborhood interior point method whose running time is the wide-neighborhood straight line complexity times current matrix multiplication time, improving in essence a factor $n$ over the algorithm by Allamigeon, Dadush, Loho, Natura, and Végh (SIAM J. Comput. 2025). The algorithm can be seen as a boosted version of the robust interior point methods of Cohen, Lee and Song (JACM 2021) and van den Brand (SODA 2020) that can reduce the gap by a polynomial factor in current matrix multiplication time. Our algorithm is also able to traverse any near-linear segments of the central path in current matrix multiplication time, independently of the length of the segment.
Our second main result focuses on interior point methods that stay in the narrow $\ell_2$-neighborhood. We give a much stronger analysis of the $\ell_2$-trust region interior point method introduced by Lan, Monteiro and Tsuchiya (SIAM J. Optim. 2009), showing that it is approximately instance optimal in this neighborhood: the number of iterations is within a constant factor of the lower bound.
A main ingredient in both methods are trust region subroutines with $\ell_\infty$ and $\ell_2$-constraints, respectively. We develop fast and strongly polynomial algorithms for solving both these problems to high accuracy. In the $\ell_2$-setting, this answers an open question by Lan, Monteiro and Tsuchiya.

We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor $O(n^{1.5}\log n)$ for an $n$-variable linear program in standard form.
The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in $O(2^{n} n^{1.5}\log(n))$ iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.

We study the circuit diameter of polyhedra and establish that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0 \le x \le u\}$ is bounded by $O(m \min\{m, n-m\} \log(m + \kappa_A) + n \log n)$, where $\kappa_A$ measures the circuit imbalance of the constraint matrix. This bound becomes strongly polynomial under certain conditions on matrix entries. Additionally, we develop circuit augmentation algorithms for linear programs using minimum-ratio circuit cancelling, solving LPs in $O(mn^2 \log(n + \kappa_A))$ augmentation steps.

We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Megiddo (SICOMP '83) and Végh (MOR '17) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL '04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices. Our approach is black-box and can be applied to any log-barrier path following method.

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $\mathrm{max} \\; c^\top x : Ax = b, x \geq 0, A \in \mathbb{R}^{m \times n}$, Vavasis and Ye developed a primal-dual interior point method using a _layered least squares_ (LLS) step, and showed that $O(n^{3.5} \log (\bar\chi_A+n))$ iterations suffice to solve (LP) exactly, where $\bar{\chi}_A$ is a condition measure controlling the size of solutions to linear systems related to $A$. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of $A$ and $c$, asked whether there exists an LP algorithm depending instead on the measure $\bar{\chi}^{\ast}_A$, defined as the minimum $\bar{\chi}_{AD}$ value achievable by a column rescaling $AD$ of $A$, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $O(m^2 n^2 + n^3)$ time algorithm which works on the linear matroid of $A$ to compute a nearly optimal diagonal rescaling $D$ satisfying $\bar{\chi}_{AD} \le n(\bar{\chi}^\ast)^3$. This algorithm also allows us to approximate the value of $\bar{\chi}_A$ up to a factor $n(\bar{\chi}^\ast)^2$. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NP-hardness for approximating $\bar{\chi}_A$ to within $2^{\mathrm{poly}(\mathrm{rank}(A))}$. The key insight for our algorithm is to work with ratios $g_i/g_j$ of circuits of $A$ — i.e., minimal linear dependencies $Ag=0$ — which allow us to approximate the value of $\bar{\chi}_A^\ast$ by a maximum geometric mean cycle computation in what we call the circuit ratio digraph of $A$. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $O(n^{2.5} \log(n) \log (\bar{\chi}^\ast_A+n))$ iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $n/\log n$ improvement on the iteration complexity bound of the original Vavasis-Ye algorithm.

We present an accelerated interior-point method for sum-of-squares polynomial optimization achieving $\tilde{O}(LU^{1.87})$ runtime, which represents polynomial improvement over existing semidefinite programming solvers. Our key contribution involves a dynamic data structure for maintaining the inverse of the Hessian of the SOS barrier function that handles multivariate cases by managing spectral approximations to low-rank perturbations of Hadamard products.

We propose a new comprehensive solution to global interconnect optimization. Traditional buffering algorithms mostly insert repeaters on a net-by-net basis based on slacks and possibly guided by global wires. We show how to integrate routing congestion, placement congestion, global timing constraints, power consumption, and additional constraints into a single resource sharing formulation. Our core contribution involves a new buffered routing subroutine that computes optimized routes based on Lagrangean resource prices. Our algorithm achieves practical efficiency, demonstrating improvements in timing while reducing netlength and power consumption on 7nm microprocessor designs.

We present an accelerated, or 'look-ahead' version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. We use the Bregman divergence as a potential function combined with combinatorial arguments to achieve improved convergence bounds. For linear fractional optimization, we improve the convergence to $O(m \log m)$ iterations from $O(m^2 \log m)$. For two-variable inequality systems, we obtain a strongly polynomial label-correcting algorithm running in $O(mn)$ iterations, with specialized $O(m + n \log n)$ performance for Markov Decision Processes. We also provide a simplified variant of parametric submodular function minimization.

We study properties and applications of various circuit imbalance measures associated with linear spaces. These measures describe possible ratios between nonzero entries of support-minimal nonzero vectors of the space. The fractional circuit imbalance measure functions as a key parameter in linear programming, while two integer variants help characterize polyhedra integrality properties. We survey both classical and recent applications, including linear programming algorithms dependent on constraint matrices and circuit augmentation methods. Additionally, we establish novel bounds on polyhedra diameter using the fractional circuit imbalance measure.

We introduce an optimization problem focused on selecting $k$ points in $[0,1]^n$ to minimize the expected cost of the cheapest point that dominates a random point from $[0,1]^n$. We demonstrate the problem is NP-hard for $k=2$ with binary coordinates, while providing an FPTAS (fully polynomial-time approximation scheme) for fixed $k$ under reasonable distributional assumptions. The work has applications where predefined products with specific feature subsets fulfill customer requests.

We extend Tardos's framework for linear programming by replacing exact LP solves with approximate ones. We demonstrate that inverse polynomial accuracy in $n$ and $\log \bar\chi_A$ suffices to solve any LP exactly. By incorporating van den Brand's recent algorithm, we achieve $O(mn^{\omega+1} \log(n)\log(\bar\chi_A+n))$ arithmetic operations, surpassing the Vavasis-Ye interior point method. Our approach combines approximate LP solutions using constructive proximity theorems to maintain low accuracy requirements while computing exact primal and dual solutions.